Summary: We define a \(\mathbb{Z}/2\)-equivariant real algebraic \(K\)-theory spectrum \(\mathrm{KR}(A)\), for every \(\mathbb{Z}/2\)-equivariant spectrum \(A\) equipped with a compatible multiplicative structure. This construction extends the real algebraic \(K\)-theory of L. Hesselholt and I. Madsen [Real algebraic \(K\)-theory. Preprint] for discrete rings, and the Hermitian \(K\)-theory of Burghelea and Fiedorowicz [1985] for simplicial rings. It supports a trace map of \(\mathbb{Z}/2\)-spectra \(\mathrm{tr}:\mathrm{KR}(A)\to\mathrm{THR}(A)\) to the real topological Hochschild homology spectrum, which extends the \(K\)-theoretic trace of M. Bökstedt et al. [Invent. Math. 111, No. 3, 465–539 (1993; Zbl 0804.55004)].
We show that the trace provides a splitting of the real \(K\)-theory of the spherical group-ring. We use the splitting induced on the geometric fixed points of \(\mathrm{KR}\), which we regard as an \(L\)-theory of \(\mathbb{Z}/2\)-equivariant ring spectra, to give a purely homotopy theoretic reformulation of the Novikov conjecture on the homotopy invariance of the higher signatures, in terms of the module structure of the rational \(L\)-theory of the “Burnside group-ring”.